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Properties

Label 4-2910e2-1.1-c1e2-0-1
Degree $4$
Conductor $8468100$
Sign $1$
Analytic cond. $539.933$
Root an. cond. $4.82042$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 2·3-s + 3·4-s − 2·5-s + 4·6-s − 7-s − 4·8-s + 3·9-s + 4·10-s − 3·11-s − 6·12-s − 6·13-s + 2·14-s + 4·15-s + 5·16-s + 2·17-s − 6·18-s − 13·19-s − 6·20-s + 2·21-s + 6·22-s + 10·23-s + 8·24-s + 3·25-s + 12·26-s − 4·27-s − 3·28-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 3/2·4-s − 0.894·5-s + 1.63·6-s − 0.377·7-s − 1.41·8-s + 9-s + 1.26·10-s − 0.904·11-s − 1.73·12-s − 1.66·13-s + 0.534·14-s + 1.03·15-s + 5/4·16-s + 0.485·17-s − 1.41·18-s − 2.98·19-s − 1.34·20-s + 0.436·21-s + 1.27·22-s + 2.08·23-s + 1.63·24-s + 3/5·25-s + 2.35·26-s − 0.769·27-s − 0.566·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8468100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8468100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8468100\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 97^{2}\)
Sign: $1$
Analytic conductor: \(539.933\)
Root analytic conductor: \(4.82042\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 8468100,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3027512253\)
\(L(\frac12)\) \(\approx\) \(0.3027512253\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$Isogeny Class over $\mathbf{F}_p$
bad2$C_1$ \( ( 1 + T )^{2} \)
3$C_1$ \( ( 1 + T )^{2} \)
5$C_1$ \( ( 1 + T )^{2} \)
97$C_1$ \( ( 1 - T )^{2} \)
good7$D_{4}$ \( 1 + T + 11 T^{2} + p T^{3} + p^{2} T^{4} \) 2.7.b_l
11$D_{4}$ \( 1 + 3 T + 21 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.11.d_v
13$D_{4}$ \( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.13.g_w
17$D_{4}$ \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.17.ac_w
19$D_{4}$ \( 1 + 13 T + 77 T^{2} + 13 p T^{3} + p^{2} T^{4} \) 2.19.n_cz
23$D_{4}$ \( 1 - 10 T + 58 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.23.ak_cg
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.29.m_dq
31$D_{4}$ \( 1 + 3 T + 35 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.31.d_bj
37$D_{4}$ \( 1 - 2 T + 62 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.37.ac_ck
41$D_{4}$ \( 1 - 9 T + 73 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.41.aj_cv
43$D_{4}$ \( 1 - 3 T + 7 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.43.ad_h
47$D_{4}$ \( 1 - 3 T + 15 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.47.ad_p
53$D_{4}$ \( 1 - 14 T + 142 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.53.ao_fm
59$D_{4}$ \( 1 + 6 T + 10 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.59.g_k
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.61.u_io
67$D_{4}$ \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.67.ac_s
71$D_{4}$ \( 1 + 3 T + 141 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.71.d_fl
73$D_{4}$ \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.73.ac_be
79$D_{4}$ \( 1 + T + 129 T^{2} + p T^{3} + p^{2} T^{4} \) 2.79.b_ez
83$D_{4}$ \( 1 - 11 T + 115 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.83.al_el
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.89.au_ks
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.001015311048471539690901689222, −8.760236371839549448857590275000, −7.945292864233655745886079405732, −7.84722916507979013940776127598, −7.37678514597402796995861784903, −7.34823898596759458318770136201, −6.68291113505211941559356149524, −6.58517821011124742711875746612, −5.81106005679289607209481650764, −5.79772687551498144233572786194, −5.09199322028416445714029597901, −4.73780283238531179851188727437, −4.32840275785903527206007771292, −3.85911119527993795424055102628, −3.09678422179390405558697398975, −2.81242964761820842716326544324, −1.97587312641184567253654675260, −1.90659891365014189574866505353, −0.54946118561905009729966069055, −0.45919929757811695271335204792, 0.45919929757811695271335204792, 0.54946118561905009729966069055, 1.90659891365014189574866505353, 1.97587312641184567253654675260, 2.81242964761820842716326544324, 3.09678422179390405558697398975, 3.85911119527993795424055102628, 4.32840275785903527206007771292, 4.73780283238531179851188727437, 5.09199322028416445714029597901, 5.79772687551498144233572786194, 5.81106005679289607209481650764, 6.58517821011124742711875746612, 6.68291113505211941559356149524, 7.34823898596759458318770136201, 7.37678514597402796995861784903, 7.84722916507979013940776127598, 7.945292864233655745886079405732, 8.760236371839549448857590275000, 9.001015311048471539690901689222

Graph of the $Z$-function along the critical line